Bootstrap for the Sample Mean and for U-Statistics of Mixing and Near Epoch Dependent Processes

TL;DR

This paper proves the strong consistency of nonoverlapping block bootstrap methods for sample means and U-statistics in complex dependent processes like ARMA, GARCH, and chaotic systems, extending bootstrap validity beyond strongly mixing data.

Contribution

It introduces bootstrap validity results for nonlinear statistics of near epoch dependent processes, including U-statistics, in a broader class of dependent data.

Findings

Bootstrap is consistent for sample mean in near epoch dependent processes.

Bootstrap is consistent for U-statistics like Gini's mean difference.

Results apply to processes such as ARMA, GARCH, and chaotic systems.

Abstract

The validity of various bootstrapping methods has been proved for the sample mean of strongly mixing data. But in many applications, there appear nonlinear statistics of processes that are not strongly mixing. We investigate the nonoverlapping block bootstrap sequences which are near epoch dependent on strong mixing or absolutely regular processes. This includes ARMA and GARCH-processes as well as data from chaotic dynamical systems. We establish the strong consistency of the bootstrap distribution estimator not only for the sample mean, but also for U-statistics, which include examples as Gini's mean difference or the chi^2-test statistic.